L2-Approximation Orders of Principal Shift-Invariant Spaces Generated by a RadialBasis Function

摘要

Approximations from the L2-closure S of the finite linear combinations of theshifts of a radial basis function are considered, and a thorough analysis of the least-squares approximation orders from such spaces is provided. The results apply to polyharmonic splines, multiquadrics, the Gaussian kernel and other functions, and include the derivation of spectral orders. For stationary refinements it is shown that the saturation class is trivial, i.e., no non-zero function in the underlying Sobolev space can be approximated to a better rate. The approach makes essential use of recent results of de Boor, DeVore and the author.